20. If a variable circle touch a given circle and a given line, the chord of contact passes through a given point.
21. If A, B, C be three points in the circumference of a circle, and D, E the middle points of the arcs AB, AC; then if the line DE intersect the chords AB, AC in the points F, G, AF is equal to AG.
22. Given two circles, O, O′; then if any secant cut O in the points B, C, and O′ in the points B′, C′, and another secant cuts them in the points D, E; D′, E′ respectively; the four chords BD, CE, B′D′, C′E′ form a cyclic quadrilateral.
23. If a cyclic quadrilateral be such that a circle can be inscribed in it, the lines joining the points of contact are perpendicular to each other.
24. If through the point of intersection of the diagonals of a cyclic quadrilateral the minimum chord be drawn, that point will bisect the part of the chord between the opposite sides of the quadrilateral.
25. Given the base of a triangle, the vertical angle, and either the internal or the external bisector at the vertical angle; construct it.
26. If through the middle point A of a given arc BAC we draw any chord AD, cutting BC in E, the rectangle AD.AE is constant.
27. The four circles circumscribing the four triangles formed by any four lines pass through a common point.
28. If X, Y , Z be any three points on the three sides of a triangle ABC, the three circles about the triangles Y AZ, ZBX, XCY pass through a common point.
29. If the position of the common point in the last question be given, the three angles of the triangle XY Z are given, and conversely.